12. donations emily has 20 collectible dolls from different countries that she will donate. if she selects…

Explanation:

Step1: Identify the number of favorable and total outcomes

We are dealing with combinations here. The total number of ways to choose 10 dolls out of 20 is given by the combination formula ( C(n, k)=\frac{n!}{k!(n - k)!} ), where ( n = 20 ) and ( k=10 ). The number of favorable outcomes is 1 (since we want a specific set of 10 dolls).

Step2: Calculate the total number of combinations

First, calculate ( C(20,10)=\frac{20!}{10!(20 - 10)!}=\frac{20!}{10!×10!} ) We know that ( 20! = 20\times19\times18\times17\times16\times15\times14\times13\times12\times11\times10! ) So, ( C(20,10)=\frac{20\times19\times18\times17\times16\times15\times14\times13\times12\times11\times10!}{10!×10!}=\frac{20\times19\times18\times17\times16\times15\times14\times13\times12\times11}{10\times9\times8\times7\times6\times5\times4\times3\times2\times1} ) Calculating the numerator: ( 20\times19 = 380 ), ( 380\times18=6840 ), ( 6840\times17 = 116280 ), ( 116280\times16=1860480 ), ( 1860480\times15 = 27907200 ), ( 27907200\times14=390700800 ), ( 390700800\times13 = 5079110400 ), ( 5079110400\times12=60949324800 ), ( 60949324800\times11 = 670442572800 ) Calculating the denominator: ( 10\times9 = 90 ), ( 90\times8 = 720 ), ( 720\times7=5040 ), ( 5040\times6 = 30240 ), ( 30240\times5=151200 ), ( 151200\times4 = 604800 ), ( 604800\times3=1814400 ), ( 1814400\times2 = 3628800 ), ( 3628800\times1=3628800 ) Now, ( \frac{670442572800}{3628800}=184756 )

Step3: Calculate the probability

The probability ( P ) is the number of favorable outcomes divided by the number of total outcomes. The number of favorable outcomes is 1 (since there is only one specific set of 10 dolls we want). So ( P=\frac{1}{C(20,10)}=\frac{1}{184756} )

Answer:

(\frac{1}{184756})